where \(\lambda _i\)  denotes the eigenvalues of the reduced density matrix.
This quantity determines the amount of information that is stored in the quantum state describing
the system of reference, which increases with its degree of mixture, and is considered to be the
best measure of entanglement for pure states [1]}.
Apart from its connection to thermodynamics [1]}, this entropy
measure has also some properties that make it of interest in quantum statistical mechanics
and quantum information theory
[3]}, [4]}, [5]}, [6]}.
Note that, despite their differences, the linear and von Neumann entropies are
going to show a similar trend as the quantum system evolves, which is also expected in the case
of their classical counterparts if there is a proper correspondence.
Furthermore, although the linear entropy is not an approximation of the von Neumann entropy,
it can readily be seen that, by Taylor expanding the latter, the first two terms correspond
to the former:
\(S_V =- {\rm Tr} (\tilde{\rho } \ln \tilde{\rho }) =- {\rm Tr} \left[ \tilde{\rho } \sum _{k=1}^\infty (-1)^{k+1}\ \frac{(\tilde{\rho } - \mathbb {I})^k}{k}\right]= 1 - {\rm Tr} (\tilde{\rho }^2)- {\rm Tr} \left[ \tilde{\rho } \sum _{k=2}^\infty (-1)^{k+1}\ \frac{(\tilde{\rho } - \mathbb {I})^k}{k}\right] .\) 
